二疊錐與半正定錐相似程度之比較
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2009
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Abstract
二疊錐與半正定錐都是對稱錐的一種特例，它們分別在二疊錐規劃與半正定錐規劃方面扮演了重要的角色。目前已知透過二疊錐與半正定錐之間的某些關係，可以將二疊錐規劃問題可以轉化成半正定錐規劃問題，但關於它們之間的一些分析技巧還是有許多不同。例如矩陣的乘積是具有結合律的，但二疊錐的Jordan乘積卻沒有。在這篇論文中，我們試著去找出、比較一些二疊錐與半正定錐之間相同或相異的地方，希望能為以後的研究提供一些想法。
The cone of positive semidefinite matrices and second-order cone are both self-dual and special cases of symmetric cones. Each of them play an important role in semidefinite programming (SDP) and second-order cone programming (SOCP), respectively. It is known that an SOCP problem can be viewed as an SDP problem via certain relation between positive semidefinite cone and second-order cone. Nonetheless, most analysis used for dealing SDP can not carried over to SOCP due to some difference, for instance, the matrix multiplication is associative for positive semidefinite cone whereas the Jordan product is not for second-order cone. In this paper, we try to have a thorough study on the similarity and difference between these two cones which provide theoretical for further investigation of SDP and SOCP.
The cone of positive semidefinite matrices and second-order cone are both self-dual and special cases of symmetric cones. Each of them play an important role in semidefinite programming (SDP) and second-order cone programming (SOCP), respectively. It is known that an SOCP problem can be viewed as an SDP problem via certain relation between positive semidefinite cone and second-order cone. Nonetheless, most analysis used for dealing SDP can not carried over to SOCP due to some difference, for instance, the matrix multiplication is associative for positive semidefinite cone whereas the Jordan product is not for second-order cone. In this paper, we try to have a thorough study on the similarity and difference between these two cones which provide theoretical for further investigation of SDP and SOCP.
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二疊錐, 凸函數, 單調函數, 半正定錐, 分解定理, second-order cone, convex function, monotone function, positive semidefinite matrix, spectral decomposition